Question

Suppose the own price elasticity of demand for good $\mathbf{X}$is -$\mathbf{3}$, its income elasticity is $\mathbf{1}$its advertising elasticity is $\mathbf{2}$, and the cross-price elasticity of demand between it and good $\mathbf{Y}$is -$\mathbf{4}$. Determine how much the consumption of this good will change if:

a. The price of good $\mathbf{X}$decreases by 5 percent.

b. The price of good Y increases by 8 percent.

c. Advertising decreases by$\mathbf{4}$percent.

d. Income increases by$\mathbf{4}$percent

Short answer

1. If the price of Good $\mathrm{X}$decreases by$5\%$ , the consumption will increase by $15\%$.
2. If the price of Good$\mathrm{Y}$ increases by $8\%$,theconsumption will decrease by $32\%$.
3. If the advertising expenditure decreases by $4\%$,the consumption will decrease by $8\%$.
4. If the income increases by $4\%$, the consumption will increase by$4\%$ .

Step-by-step solution

Step 1: To find the consumption of Good  X when price decreases by 5%

a.

When price decreases by $5\%$,we use the following equationto determine the consumption of Good $\mathrm{X}$:

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{\mathrm{\%}{\mathrm{\Delta P}}_{\mathrm{x}}}$

The given data for${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}}.{\mathrm{P}}_{\mathrm{x}}}=-3;\mathrm{and}\mathrm{\%}{\mathrm{\Delta P}}_{\mathrm{x}}=-5.$

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{x}}}=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{\mathrm{\%}{\mathrm{\Delta P}}_{\mathrm{x}}}$

Put the value ofthe given data.

$\begin{array}{c}-3=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{-5}\\ \mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}=15\mathrm{\%}\end{array}$

Thus, if the price of Good$\mathrm{X}$ decreases by $5\%$, the consumption will increase by $15\%$.

Step 2: To find the consumption of Good X when the price increases by 8%

b.

When the price increases by $8\%$,we use the following equation to determine the consumption of Good $\mathrm{X}$.

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{y}}}=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{\mathrm{\%}{\mathrm{\Delta P}}_{\mathrm{z}}}$

The given data for${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{y}}}=-4;\mathrm{and}\mathrm{\%}{\mathrm{\Delta P}}_{\mathrm{x}}=8$ .

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},{\mathrm{P}}_{\mathrm{y}}}=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{\mathrm{\%}{\mathrm{\Delta P}}_{\mathrm{y}}}$

Put the value ofthe given data:

$\begin{array}{c}-4=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{8}\\ \mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}=-32\mathrm{\%}\end{array}$

Thus, if the price of Good$\mathrm{Y}$ increases by$8\%$ , consumption will decrease by$32\%$ .

To find the consumption of Good X  when advertising expenditure decreases by 4%

c.

If the advertising expenditure decreases by $4\%$, we use the following equation to determine the consumption of Good $\mathrm{X}$:

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{A}}=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{\mathrm{\%}\mathrm{\Delta A}}$

The given data for${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{A}}=2;\mathrm{and}\mathrm{\%}\mathrm{\Delta A}=-4$ .

$\begin{array}{c}{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{A}}=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{\mathrm{\%}\mathrm{\Delta A}}\\ \begin{array}{l}2=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{-4}\\ \mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}=-8\mathrm{\%}\end{array}\end{array}$

Thus, when advertising expenditure decreases by $4\%$, Good consumption will decrease by $8\%$.

To determine the consumption of Good X  when the income increases by 4% 

d.

If the income increases by$4\%$ , we use the following equationto determine the consumption of Good $\mathrm{X}$.

${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{M}}=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{\mathrm{\%}\mathrm{\Delta M}}$

The given data ${\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{M}}=1;\mathrm{and}\mathrm{\%}\mathrm{\Delta M}=4$.

$\begin{array}{c}{\mathrm{E}}_{{\mathrm{Q}}_{\mathrm{x}},\mathrm{M}}=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{\mathrm{\%}\mathrm{\Delta M}}\\ 1=\frac{\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}}{4}\end{array}$

$\mathrm{\%}{\mathrm{\Delta Q}}_{\mathrm{x}}^{\mathrm{d}}=4\mathrm{\%}$

Thus, when the income increases by $4\%$, Good consumption will increase by $4\%$.